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6.3 molecular orbital theory and the linear combination of atomic orbitals approximation for H2+ The goal of this section is to introduce the first of the two main methods for generating approximate electronic wave functions for molecules. The LCAO method consists of selecting sums and differences (linear combinations) of atomic orbitals to generate the best approximation to each type of molecular orbital. The approximate MO constructed in this way is then used to calculate properties of the molecule using standard equations of quantum mechanics, by hand for simple cases and by computer for larger molecules. The LCAO method is best explained with the help of a specific example, so we start with the hydrogen molecular ion H2 +. We can evaluate the success of the method by comparing its results to the exact solution described in Section 6.2, and gain confidence in applying the method to more complex molecules. The LCAO approximation is motivated by physical reasoning as follows. Consider the 1 sg MO shown in Figure 6.5. When the electron is close to nucleus A, it experiences a potential not very different from that in an isolated hydrogen atom. The ground-state MO in the region near A should therefore resemble a 1s atomic wave function 1s A. Near B, the MO should resemble 1s B. The 1s orbitals at A and B are identical; the labels are attached to emphasize the presence of two nuclei. Note that A and B are merely labels, not exponents. A simple way to construct a MO with these properties is to approximate the 1sg MO as a sum of the H 1s orbitals with adjustable coefficients. Similarly, we can approximate the other exact MOs in Figure 6.5 by forming linear combinations of the H atomic orbitals which they resemble when the electron is near A or B. The general form for an approximate MO for H2 + is a linear combination of two atomic orbitals, obtained by adding or subtracting the two: with coefficients whose values depend on RAB: - ? ? MO A AB A B AB B( ) ( )= ± C R C R s s1 1 For our purposes, it is adequate to ignore the dependence of the coefficients on RAB and evaluate them by normalizing the wave function. The coefficients CA and CB give the relative weights of the two atomic orbitals in determining the character of the MO. If CA were greater in magnitude than CB, the 1s A orbital would be more heavily weighted and the electron would be more likely to be found near nucleus A and vice versa. But because the two nuclei in the H2 + ion are identical, the electron is just as likely to be found near one nucleus as the other. Therefore, the magnitudes of CA and CB must be equal, so either CA??5? ? CB or CA??5??2CB. For both these choices ( ) MO 2 does not change if the A and B nuclei are interchanged. To maintain the distinctions among the various orbitals, we will use the following notation in describing the LCAO approximation. Atomic orbitals will be represented by and molecular orbitals by s or p. Generic wave functions will be represented by . Occasionally, will represent some special wave function, in which case appropriate subscripts will be attached. LACO molecular orbitals for H2+ Proceeding as described above, we construct approximate molecular orbitals for the exact 1 g and 1 u ? MOs in Figure 6.5: 1 1 1 1 - - ? ? g g s g s s C ≈ = + [ ] A B [6.2a] 1 1 1 1 - - ? ? u u s u s s C? ≈ = ? ? [ ] A B [6.2b] where Cg and Cu are chosen to ensure that the total probability of finding the electron somewhere is unity and we have ignored their dependence on the choice of RAB at which we have fixed the nuclei. Notice that we have introduced new symbols g s 1 and u s 1 ? for the approximate MOs not only to distinguish them from the exact MOs but also to indicate explicitly the AOs from which they were constructed. The symbols for the exact and approximate MOs are summarized in Table 6.1. To simplify the notation, henceforth we omit the dependence of g s 1 and u s 1 ? on RAB. The distribution of electron probability density is obtained by squaring each of the approximate MOs: [ ] [( ) ( ) ] - ? ? ? ? g s g s s s s C1 2 2 1 2 1 2 1 12= + + A B A B [6.3a] [ ] [( ) ( ) ] - ? ? ? ? u s u s s s s C1 2 2 1 2 1 2 1 12? = + ? A B A B [6.3b] These can be compared with the electron probability distribution for a noninteracting (n.i.) system (obtained by averaging the probabilities for HA??1??HB + and HA +??1??HB), which is - ? ? 2 3 2 1 2 12n.i.A B = + C s s [( ) ( ) ] [6.4] To describe the noninteracting system as one electron distributed over two possible sites, we set C3 2??5??0.5. The interpretation of these approximate MOs in relation to the noninteracting system is best explained graphically. The plots of these various wave functions (left side) and their squares (right side) are shown in Figure 6.7. Compared with the noninteracting pair of atoms, the system described by the approximate MO g s 1 shows increased electron density between the nuclei and is therefore a bonding orbital as defined in Section 6.2. By contrast, the approximate MO u s 1 ? shows reduced probability for finding an electron between the nuclei and so is an antibonding orbital. Note in Figure 6.7 that u s 1 ? has a node between the nuclei and is antisymmetric with respect to inversion through the molecular center (see Section 6.2). Comparing Figure 6.7 with Figure 6.5 shows that the LCAO method has reproduced qualitatively the probability density in the first two exact wave functions for H2+. energy of h1 2 in the lcao approximation To complete the demonstration that g s 1 and u s 1 ? are bonding and antibonding MOs respectively, we examine the effective potential energy curve for the H2 + ion with its electron in each of these approximate MOs. Figure 6.8 shows the effective potential energy of the H2 + ion in the LCAO approximation for the g s 1 and u s 1 ? MOs. The force between the nuclei in the antibonding state is everywhere repulsive, but in the bonding state the nuclei are attracted to each other and form a bound state at the distance corresponding to the lowest potential energy. The energy minimum of the potential at Re is called De, the bond dissociation energy, the energy required to dissociate the molecular ion into a separated proton and a hydrogen atom. At Re, where the effective potential energy has its minimum value, the attractive and repulsive forces between the nuclei balance exactly. The equilibrium bond length of the molecule is determined by the competition between attractive forces, which originate in electron-nuclear interactions, and repulsive forces, which originate in nuclear–nuclear interactions. This is the sense in which the electrons provide an attractive force that holds the nuclei to their equilibrium positions that define the structure and geometry of a molecule. How well does the LCAO approximation describe the effective potential energy curve in H2 +? We compare the exact and LCAO results in Figure 6.9, where the zero of energy at infinite separation is again taken to be the separated species H and H1. The energy in g s 1 has a minimum at RAB??5??1.32 ?, and the predicted energy to dissociate the ion to H and H1 is D??5??1.76 eV. These results compare reasonably well to the experimentally measured values RAB??5??1.06 ? and D0??5??2.79 eV, which were also obtained from the exact solution in Section 6.2. Let’s put these results of the LCAO approximation in perspective. The results in Figure 6.7 and Figure 6.8 were obtained by working out the details of the approximation expressed in Equation 6.2ab. These LCAO results have captured qualitatively the results of the exact calculation. Therefore we can apply the LCAO method in other more complex cases and be confident we have included the qualitative essential features of bond formation. And, we can always improve the results by following up with a self-consistent computer calculation that produces optimized MOs. We will give some examples later in the chapter to illustrate how advanced calculations flow very easily from the simple LCAO theory introduced here. The energy-level diagram within the LCAO approximation is given by a correlation diagram (Fig. 6.10), which shows that two 1s atomic orbitals have been combined to give a g s 1 MO with energy lower than the atomic orbitals and a u s 1 ? MO with higher energy than the atomic orbitals. This diagram is a purely qualitative representation of the same information contained in Figure 6.8. The actual energy level values will depend on the distance between the fixed nuclei (as shown in Figure 6.8) and must be determined from calculations. Even without the results shown in Figure 6.8, we would know that an electron in an antibonding orbital has higher energy than one in a bonding orbital because the antibonding orbital has a node. Consequently, in the ground state of H2 +, the electron occupies the g s 1 molecular orbital. By forming the bond in the molecular ion, the total system of two hydrogen nuclei and one electron becomes more stable than the separated atoms by the energy difference 2DE shown in Figure 6.10. 6.4 homonuclear diatomic molecules: first-Period atoms We can combine the LCAO method for H2 + with an aufbau principle, analogous to that developed for atoms, to describe the electron configurations of more complex molecules. Electrons available from the two atoms are “fed” into the molecular orbitals, starting with the MO of lowest energy. At most, two electrons can occupy each molecular orbital. The ground-state H2 molecule, therefore, accommodates two electrons with opposite spins in a g s 1 bonding molecular orbital, as shown in Figure 6.11. The diatomic molecule is more stable than the isolated atoms by the energy difference 2DE. The value of 2DE here is different from that in Figure 6.10 because the present case involves the effects of electron–electron repulsion. The LCAO approximation can be applied in this same way to He2 + and He2, with one change. The MOs must be generated as linear combinations of He 1s atomic orbitals, not H 1s orbitals. The reason is that when the electrons in He2 + and He2 approach close to one of the nuclei, they experience a potential much closer to that in a He atom than in a H atom. Therefore, the equations for the MOs are - ? ? g s g He s He s C1 1 1 = + [ ] A B [6.5a] - ? ? g s u He s He s C1 1 1? = ? [ ] A B [6.5b] We rewrite Equations 6.5a and 6.5b using a simpler notation, which we adopt for the remainder of the text. g s gC s s 1 1 1 = + [ ] A B [6.6a] g s gC s s 1 1 1* [ ] = ? A B [6.6b] In these equations the symbol for the atomic wave function has been dropped and the atomic orbitals are identified by their hydrogenic labels 1s, 2s, 2p, etc. The superscripts A and B will be used to identify particular atoms of the same element in bonds formed from the same element in homonuclear diatomics. These helium MOs have the same general shapes as those shown in Figure 6.7 for the MOs constructed from H 1s. They lead to potential energy curves as shown in Figure 6.8 and a correlation diagram similar to Figure 6.11. Quantitative calculations of electron density and energy (these calculations are not performed in this book) would produce different values for the two sets of MOs in Equations 6.2a and 6.2b and Equations 6.5a and 6.5b. Keep in mind that we construct the MOs as combinations of all the atomic orbitals required to accommodate the electrons in the ground states of the atoms forming the molecule. This set of atomic orbitals is called the minimum basis set for that specific molecule. Therefore, quantitative calculations for each molecule are influenced by the detailed properties of the atoms in the molecule. Because He2 + and He2 have more than two electrons, the aufbau principle requires them to have some electrons in the u s 1? antibonding orbital. The electrons in the antibonding orbital contribute a destabilization energy in the amount 1DE relative to the separated He atoms, as shown in Figure 6.12. This effect competes with the stabilization energy of 2DE that arises from the electrons in the g s 1 bonding orbital, giving a weak bond in He2 + and no stable bond in He2. The general features of covalent bonding in the LCAO picture can be summarized as follows. Covalent bond formation arises from the presence of electrons (most often electron pairs with opposite spins) in bonding MOs. The average electron density between the nuclei is greater than between the non-interacting atoms, and tends to pull them together. Electrons in an antibonding MO tend to force the nuclei apart, reducing the bond strength. This competition is described by the bond order, defined as follows: bond order??5???1 2 (number of electrons in bonding molecular orbitals 2 number of electrons in antibonding molecular orbitals) In the LCAO molecular orbital description, the H2 molecule in its ground state has a pair of electrons in a bonding molecular orbital and thus a single bond (that is, its bond order is one). Later on, as we describe more complex diatomic molecules in the LCAO approximation, we will see bond orders greater than one. This quantum mechanical definition of bond order generalizes the concept first developed in the Lewis theory of chemical bonding, that a shared pair of electrons corresponds to a single bond, two shared pairs to a double bond, and so forth. Table 6.2 lists the molecular orbital configurations of homonuclear diatomic molecules and molecular ions made from first-period elements. These configurations are simply the occupied molecular orbitals in order of increasing energy, together with the number of electrons in each orbital. Higher bond order corresponds to greater bond dissociation energies and shorter bond lengths. The species He2 has bond order 0 and does not form a true chemical bond. The preceding paragraphs have illustrated the LCAO approximation with specific examples and shown how the character of the chemical bond is determined by the difference in the number of electrons in bonding and antibonding MOs. The systematic procedure for applying the LCAO approximation to define the MOs for any diatomic molecule consists of three steps: 1. Form linear combinations of the minimum basis set of AOs (all of the AOs occupied in the ground state of each atom in the molecule) to generate MOs. The total number of MOs formed in this way must equal the number of AOs used. 2. Arrange the MOs in order from lowest to highest energy. 3. Put in electrons (at most two electrons per MO), starting from the orbital of lowest energy. Apply Hund’s rules when appropriate. 6.5 homonuclear diatomic molecules: Second-Period atoms Now that we have some experience with the LCAO method, it is useful to collect several important insights before we proceed to more complicated molecular examples. The LCAO method extends to molecules the description developed for many-electron atoms in Section 5.2. We wrote the electron configuration for a many-electron atom by placing electrons in a set of single-particle AOs according to the Pauli exclusion principle. Here we will see how to write the electron configuration for a molecule by placing electrons in a set of single-particle MOs, guided by the Pauli exclusion principle. How do we obtain the single-electron molecular orbitals? The LCAO method was invented to construct approximate molecular orbitals directly from the Hartree atomic orbitals for the specific atoms in the molecule, guided by molecular symmetry and chemical intuition. We start with the Hartree AOs (see Section 5.2) because they already include the effects of the atomic number Z and of shielding on the size and energy of the AOs in each atom. We thus have a quick check on, for example, the relative energies of the 2s and 2p AOs in a pair of different atoms, and from this we can quickly see the extent to which electrons in these AOs interact or compete to determine the size and energy of the resulting MO. The essential new feature compared to the atomic case is that the (multi-center) approximate MOs are spread around all the nuclei in the molecule, so the electron density is delocalized over the entire molecule. The approximate MOs therefore differ considerably from the (single-center) atomic orbitals used in Section 5.2. Constructing the approximate MOs and recognizing how their electron density is distributed over the entire network of fixed nuclei are the key tools for describing molecular structure and bonding. Mastering these tools for progressively more complicated molecules is our objective in this Section and the next three after it. In order to discuss bonding with atoms in the second period and beyond, we must generate approximate MOs to accommodate electrons from AOs higher than the 1s orbitals. Let’s try to motivate these combinations by physical reasoning, as we did at the beginning of Section 6.3. Imagine that two Li atoms in their ground-state electronic configurations (1s)2(2s)1 approach each other. It is reasonable to expect that their 2s AOs will interact to produce MOs very similar in shape to the 1s bonding and anti-bonding MOs but larger in size, and that two electrons will go into this new bonding MO which we temporarily name 2s . What happens to the 1s electrons? We can use Figure 5.25 to estimate that the energy of the Li(1s) AO is about 3.8 Ry (or 52 eV) below that of the Li(2s) AO. Because this is a very significant difference, it is unlikely that the 1s and 2s electrons will interact, and it is reasonable to suppose the 1s orbitals will combine to form 1s bonding and anti-bonding MOs as described in the previous section, independent of the 2s electrons. If our thinking is correct, then the Li2 molecule would have the electron configuration ( ) ( ) ( ) * g s u s g s 1 2 1 2 2 2 . In a similar way we would expect Be2 to have the configuration ( ) ( ) ( ) ( ) * * g s u s g s u s 1 2 1 2 2 2 2 2. In the remainder of Period 2 from B to Ne we must consider the possibility that the 2p AOs will interact with each other and also with the 2s AOs. Because the 2p AOs are not spherical, we expect the relative orientation of the two orbitals will strongly influence the formation of MOs. Qualitatively, we expect that two dumbbell shapes interacting end-to-end will produce a different result than side-to-side or end-to-side. Moreover, the difference in phase of the two lobes will strongly influence the result. Careful geometrical analysis is required to identify the MOs formed from the 2p AOs. Two conclusions from more advanced discussions of quantum theory justify the results of our physical reasoning and tell us how to construct the MOs for second-period atoms in a systematic way. 1. Two atomic orbitals contribute significantly to bond formation only if their atomic energy levels are very close to one another. Consequently, we can ignore mixing between the core-shell 1s orbitals and the valence-shell 2s and 2p orbitals. Similarly, we can ignore mixing between the 2s and 2p orbitals except in special cases to be described below. 2. Two atomic orbitals on different atoms contribute significantly to bond formation only if they overlap significantly. The term overlap is used in bonding theory to describe the extent to which orbitals on separate atoms “interact” or “inter-penetrate” as two atoms approach one another at close distance. Two orbitals overlap significantly if both have appreciable amplitudes over the same region of space. The net overlap may be positive, negative, or zero, depending on the relative amplitudes and phases of orbitals involved in the overlap region. Bonding molecular orbitals arise from positive net overlap (constructive interference between the wave functions for atomic orbitals), antibonding molecular orbitals result from negative net overlap (destructive interference between the wave functions for atomic orbitals). Nonbonding molecular orbitals can arise in two ways: a) there may be negligible overlap of the atomic orbitals because the separation between the nuclei is greater than the spatial extent of the orbitals or b) regions of positive overlap cancel regions of negative overlap to give zero net overlap. We give specific examples in the following paragraphs. For s orbitals it is rather easy to guess the degree of overlap; the closer the nuclei, the greater the overlap. If the wave functions have the same phase, the overlap is positive; if they have opposite phases, the overlap is zero. For more complex cases, the overlap between participating atomic orbitals depends strongly on both the symmetry of the arrangement of the nuclei and on the phases of the orbitals. If the two orbitals are shaped so that neither has substantial amplitude in the region of interest, then their overlap is negligible. However, if they both have significant amplitude in the region of interest, it is important to know whether regions of positive overlap (where the two orbitals have the same phase) are canceled by regions of negative overlap (where the two orbitals have opposite phases). Such cancellation leads to negligible or zero overlap between the orbitals. Qualitative sketches that illustrate significant or negligible overlap in several common cases are shown in Figure 6.13. In particular, constructive interference and overlap between s and p orbitals is significant only in the case where an s orbital approaches a p orbital “end-on.” The phase of the p orbital lobe pointing toward the s orbital must be the same as that of the s orbital. You should review the “sizes and shapes” of hydrogenic orbitals discussed in Section 5.1 and depicted in Figure 5.4. A great deal of qualitative insight into the construction of molecular orbitals can be gleaned from these considerations. The two conclusions stated above justify the following approximate LCAO MOs for the second-period homonuclear diatomic molecules. As with the first-period atoms, we use the new labels in Table 6.1 to distinguish the approximate MOs from the exact H2 + MOs in Figure 6.5 and to indicate their atomic parentage. We combine the 2s atomic orbitals of the two atoms in the same fashion as 1s orbitals, giving a g s 2 bonding orbital and a u s 2 ? antibonding orbital: g s gC s s 2 2 2 = + [ ] A B [6.7a] u s uC s s 2 2 2? = ? [ ] A B [6.7b] The choice of appropriate combinations of the 2p orbitals is guided by the overlap arguments and by recalling that the bond axis is the z-axis. The 2p orbitals form different MOs depending on whether they are parallel or perpendicular to the internuclear (bond) axis. Consider first the 2pz orbitals, which can be used to form two different kinds of s orbitals. If the relative phases of the pz orbitals are the same so they interfere constructively in the internuclear region, then a bonding sg2p orbital is formed. Conversely, if lobes of opposite phase overlap, they form an antibonding MO labeled u pz 2 ? . These MOs are shown in Figure 6.14. g p g z z z C p p2 2 2 = ? [ ] A B [6.8a] u p u z z x C p p2 2 2? = + [ ] A B [6.8b] The bonding MO shows increased electron density between the nuclei, whereas the antibonding MO has a node.
The two 2px orbitals, whose lobes are oriented perpendicular to the bond axis, can overlap “side by side” to form a bonding MO denoted u px 2 and an antibonding MO denoted g px 2 ? . The overlap leading to these MOs is shown in Figure 6.15. u p u x x x C p p2 2 2 = + [ ] A B [6.9a] g p g x x x C p p2 2 2? = ? [ ] A B [6.9b] These orbitals have a nodal plane that contains the internuclear axis (in this case, the y-z plane). They are designated by p to signify their origin in AOs that have one unit of angular momentum along the internuclear axis. The p MOs do not have cylindrical symmetry about the internuclear axis. In the same way, u py 2 and g py 2 ? orbitals can be formed from the 2py atomic orbitals. Their lobes project above and below the x-z nodal plane, which is the plane of the page in Figure 6.15. u p u y y y C p p2 2 2 = + [ ] A B [6.10a] g p g y y y C p p2 2 2? = ? [ ] A B [6.10b] Like the 2p AOs from which they are constructed, the 2p MOs are degener ate: u px 2 and u py 2 have the same energy, and g px 2 ? and g py 2 ? have the same energy. We expect u px 2 and u py 2 to be less effective than g pz 2 as bonding orbitals, because the overlap in the p orbitals occurs off the internuclear axis and therefore has less tendency to increase the electron density between the nuclei and to pull them closer together. The most important point to understand in constructing MOs and predicting their behavior by the overlap argument is that the relative phases of the two AOs determine whether the resulting MO is bonding or antibonding. Bonding orbitals form through the overlap of wave functions with the same phase, by constructive interference of “electron waves”; antibonding orbitals form through the overlap of wave functions with opposite phase, by destructive interference of “electron waves.” By considering the symmetry and relative energies of the participating AOs, we have generated a total of 8 approximate LCAO MOs to accommodate the electrons in the 2s and 2p AOs. Including the two MOs originating from the 1s AOs, we now have a total of 10 approximate MOs which can describe diatomic molecules containing up to 20 electrons, from H2 to Ne2. Each of these is a simple sum or difference of two AOs. The next step is to determine the energy ordering of the MOs. In general, that step requires a calculation, as we did for the first-period diatomics in Figure 6.9 and Figure 6.10. The results for Li2 through F2 are shown in Figure 6.16. The electrons for each molecule have been placed in MOs according to the aufbau principle. We show only the MOs formed from the 2s and 2p orbitals. In second-period diatomic molecules, the 1s orbitals of the two atoms barely overlap. Because the g s 1 bonding and u s 1 ? antibonding orbitals are both doubly occupied, they have little net effect on bonding properties and need not be considered. There are two different energy orderings for diatomic molecules formed from second-period elements. The first (Fig. 6.17a) applies to the molecules with atoms Li through N (that is, the first part of the period) and their positive and negative ions. The second (Fig. 6.17b) applies to the later elements, O, F, and Ne, and their positive and negative ions. This difference is based on experimental measurements of molecular orbital energies by photoelectron spectroscopy (see Section 6.7) summarized in Figure 6.16. While the energy of the u px y 2 ( ) molecular orbital remains nearly constant as we move across Period 2, the energy of the g pz 2 molecular orbital lies above u px y 2 ( ) in the first part of the period and falls below it for the later elements O, F, and Ne. This behavior is borne out by advanced calculations of the energies for these two molecular orbitals. A simplified physical interpretation relates this result to the extent to which the 2s and 2p atomic orbitals can mix while contributing to the molecular orbital. The 2s AOs have the right symmetry to mix with 2p AOs to form a s molecular orbital directed along the z-axis. In the first part of Period 2 the 2s and 2p are sufficiently close in energy that mixing occurs and increases the MO energy. In the latter part the energy separation of 2p and 2s AOs is too great for mixing to occur. Because the 2s and 2px and 2py AOs do not have the right symmetry to mix and form a p molecular orbital, the energy of the u px y 2 ( ) MO remains nearly constant across the period even when the 2s and 2p AOs are close together in energy. An important prediction comes from the correlation diagrams in Figure 6.17. Hund’s rules require that, in the ground state, the electrons occupy different orbitals and have parallel spins, so B2 and O2 are predicted to be paramagnetic. This paramagnetism is exactly what is found experimentally (Fig. 6.18). In contrast, in the Lewis electron dot diagram for O2, O O all the electrons appear to be paired. Moreover, the extremely reactive nature of molecular oxygen can be rationalized as resulting from the readiness of the two p* electrons, unpaired and in different regions of space, to find additional bonding partners in other molecules. The electron configurations in Figure 6.16 allow us to calculate the bond order for each molecule and correlate it with other properties of the molecules. example 6.3 Determine the ground-state electron configuration and bond order of the F2 molecule. Solution Each atom of fluorine has 7 valence electrons, so 14 electrons are placed in the molecular orbitals to represent bonding in the F2 molecule. The correlation diagram of Figure 6.17 gives the electron configuration ( ) ( ) ( ) ( , ) ( , - - - ? ? ? ? g s u s g p u p u p g p g z x y x2 2 2 2 2 2 2 2 4 2p p 22 4 pyp ) Because there are 8 valence electrons in bonding orbitals and 6 in antibonding orbitals, the bond order is bond order??5??1 2 (8 2 6)??5??1 and the F2 molecule has a single bond. related problems: 17, 18, 19, 20, 21, 22, 23, 24 Table 6.3 summarizes the properties of second-period homonuclear diatomic molecules. Note the close relationship among bond order, bond length, and bond energy and the fact that the bond orders calculated from the MOs agree completely with the results of the Lewis electron dot model. How these properties depend upon the number of electrons in the molecules is shown in Figure 6.19. The bond orders simply follow the filling of MOs in a given subshell, rising from zero to one and then falling back to zero for the first-period diatomics and also for Li2, Be2, and their ions. The bond orders move in half-integral steps if we include the molecular ions; they increase as the s orbitals are filled and decrease as the s* orbitals begin to fill. These MOs are all constructed from s orbitals and so there is no possibility for multiple bonds. Moving from Be2 through Ne2, the bond orders move again in half-integer increments from zero to three and back to zero as the p orbitals and then the p* orbitals are filled. Both bond energies and force constants are directly correlated with the bond order, whereas the bond length varies in the opposite direction. This makes perfect sense; multiple bonds between atoms should be stronger and shorter than single bonds. In summary, the simple LCAO method provides a great deal of insight into the nature of chemical bonding in homonuclear diatomic molecules and the trends in their properties that result. It is consistent with the predictions of simpler theories, like that of G. N. Lewis; but clearly more powerful and more easily generalized to problems of greater complexity. 6.6 heteronuclear diatomic moleculeS Diatomic molecules such as CO and NO, formed from atoms of two different elements, are called heteronuclear. We construct molecular orbitals for such molecules by following the procedure described earlier, with two changes. First, we use a different set of labels because heteronuclear diatomic molecules lack the inversion symmetry of homonuclear diatomic molecules. We therefore drop the g and u subscripts on the MO labels. Second, we recognize that the AOs on the participating atoms now correspond to different energies. For example, we combine the 2s atomic orbital of carbon and the 2s atomic orbital of oxygen to produce a bonding MO (without a node) 2 2 2 s C s C s = + A A B B [6.11a] and an antibonding MO (with a node) 2 2 2 s C s C sp = ′ ? ′ A A B B [6.11b] where A and B refer to the two different atoms in the molecule. In the homonuclear case we argued that CA??5??CB and ′ CA ??5?? ′ CB because the electron must have equal probability of being near each nucleus, as required by symmetry. When the two nuclei are different, this reasoning does not apply. If atom B is more electronegative than atom A, then CB??.??CA for the bonding s MO (and the electron spends more time on the electronegative atom); ′ CA ??.?? ′ CB for the higher energy s* MO and it more closely resembles a 2sA AO. Molecular orbital correlation diagrams for heteronuclear diatomics start with the energy levels of the more electronegative atom displaced downward because that atom attracts valence electrons more strongly than does the less electronegative atom. Figure 6.20 shows the diagram appropriate for many heteronuclear diatomic molecules of second-period elements (where the electronegativity difference is not too great). This diagram has been filled with the valence electrons for the ground state of the molecule BO. Another example, NO, with 11 valence electrons (5 from N, 6 from O), has the ground-state configuration ( ) ( ) ( , ) ( ) ( , ) - - ? ? - ? ? 2 2 2 2 2 2 4 2 2 2 2 1 s s p p p p p x y z x y p p With eight electrons in bonding orbitals and three in antibonding orbitals, the bond order of NO is 1 2 (8 2 3)??5??2 1 2 and it is paramagnetic. The bond energy of NO should be smaller than that of CO, which has one fewer electron but a bond order of three; experiment agrees with this prediction. We explained earlier that in homonuclear diatomics, atomic orbitals mix significantly to form molecular orbitals only if they are fairly close in energy (within 1 Ry or so) and have similar symmetries. The same reasoning is very helpful in constructing MOs for heteronuclear diatomics. For example, in the HF molecule, both the 1s and 2s orbitals of the F atom are far too low in energy to mix with the H 1s orbital. Moreover, the overlap between the H 1s and F 2s is negligible (Fig. 6.21a). The net overlap of the H 1s orbital with the 2px or 2py F orbital is zero (Fig. 6.21b) because the regions of positive and negative overlap sum to zero. This leaves only the 2pz orbital of F to mix with the H 1s orbital to give both s bonding and s* anti-bonding orbitals (see Figs. 6.21b and d). The correlation diagram for HF is shown in Figure 6.22. The 2s, 2px, and 2py orbitals of fluorine do not mix with the 1s of hydrogen and therefore remain as atomic (nonbonding) states denoted by snb and pnb. Electrons in these orbitals do not contribute significantly to chemical bonding. Because fluorine is more electronegative than hydrogen, its 2p orbitals lie below the 1s hydrogen orbital in energy. The s orbital then contains more fluorine 2pz character, and the s* orbital more closely resembles a hydrogen 1s AO. When the eight valence electrons are put in for HF, the result is the electron configuration: ( ) ( ) ( , ) - - ? ? nb nb nb 2 2 4 x y The net bond order is 1 because electrons in nonbonding AOs do not affect bond order. The electrons in the s orbital are more likely to be found near the fluorine atom than near the hydrogen, so HF has the dipole moment H1F2. If a more electropositive atom (such as Na or K) is substituted for H, the energy of its outermost s orbital will be higher than that of the H atom, because its ionization energy is lower. In this case, the s orbital will resemble a fluorine 2pz orbital even more (that is, the coefficient CF of the fluorine wave function will be close to 1, and CA for the alkali atom will be very small). In this limit the molecule can be described as having the valence electron configuration ( ) ( ) ( , , ) - which corresponds to the ionic species Na1F2 or K1F2. The magnitudes of the coefficients in the molecular orbital wave function are thus closely related to the ionic–covalent character of the bonding and to the dipole moment.
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